Discussion Topic: What Is The Domain Of The Following Logari

Discussion Topicwhat Is The Domain Of The Following Logarithmic Functi

Discussion Topic what Is The Domain Of The Following Logarithmic Functi

Discussion Topic what Is The Domain Of The Following Logarithmic Functions? 1) f(x) = log(3x - ) 2) f(x) = log(2x + ) 3) f(x) = log2(x - 2) Perform the operations: 4) log2 4 + log4 2 = 5) log 1000 + ln e5 = 6) log1/2 8 = Solve the equations: 7) x2 = log 100 + ) Solve the equation log 4 + log x = log ) Solve the equation log (x + 3) + log x = log 28 On all the exercises show your work step-by-step. No work- No credit.

Paper For Above instruction

The provided exercises focus on understanding the domain of logarithmic functions and performing operations involving logarithms and solving logarithmic equations. These are fundamental components in algebra and calculus, which require careful attention to the properties of logarithms and the constraints imposed by the domain of these functions.

Understanding the Domain of Logarithmic Functions

The domain of a logarithmic function is determined by the argument of the logarithm being positive because the logarithm of a non-positive number is undefined in the real number system. For a function f(x) = log_b (g(x)), the condition for the domain is g(x) > 0.

1) For f(x) = log(3x - ?), the argument is 3x - ?. Since the expression is incomplete, we assume the intent is similar to other functions, likely something like log(3x - a). The domain constraint would be 3x - a > 0, which simplifies to x > a/3.

2) For f(x) = log(2x + ?), again assuming a missing value, the domain constraint is 2x + ? > 0. If we presume the missing part is a constant value, say, 'b', this simplifies to x > -b/2.

3) For f(x) = log2 (x - 2), the argument must be positive, so x - 2 > 0. Therefore, the domain is x > 2.

Performing Logarithmic Operations

4) To compute log2 4 + log4 2, we can convert both to a common base or use change-of-base formula. Recognizing that log2 4 = 2 because 22 = 4. Also, log4 2 = 1/2 because 41/2 = 2. Adding these, 2 + 0.5 = 2.5.

5) For log 1000 + ln e5, assuming base 10 for the first log, log 1000 = 3 since 103 = 1000. And, ln e5 = 5. The sum is 3 + 5 = 8.

6) For log1/2 8, recall that log1/2 8 is the power to which 1/2 must be raised to produce 8. Since (1/2)^n = 8 is equivalent to 2-n = 8. Because 8 = 23, we have 2-n = 23, which implies -n = 3, so n = -3. Therefore, log1/2 8 = -3.

Solving Logarithmic Equations

7) For x2 = log 100, first compute log 100. Assuming base 10, log 100 = 2. The equation becomes x2 = 2. Taking square roots yields x = ±√2.

8) To solve log 4 + log x = log, the expression is incomplete, but likely the intent is log 4 + log x = log y. Using the logarithm product rule: log (4x) = log y. Setting the arguments equal: 4x = y. Without the value for y, we cannot proceed further, but if y is specified, solving for x is straightforward.

9) For log (x + 3) + log x = log 28, application of the product rule gives log [(x + 3) x] = log 28. This simplifies to (x + 3) x = 28. Expanding, x2 + 3x = 28. Rearranged as x2 + 3x - 28 = 0. Factoring or using quadratic formula: x = [-3 ± √(9 + 112)] / 2 = [-3 ± √121] / 2. Hence, x = [-3 + 11]/2 = 4 or x = [-3 - 11]/2 = -7. Considering the domain constraints, we discard the negative solution where it makes any argument of the logarithm non-positive. Since x + 3 > 0 implies x > -3, the solution x = 4 is valid, whereas x = -7 is invalid because it would make x + 3 = -4.

Conclusion

These exercises highlight critical steps in understanding the properties of logarithms, including domain restrictions, logarithmic identities, and solving logarithmic equations. Mastery of these skills is essential for progressing in algebra, calculus, and other advanced mathematics fields.

References

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  • National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. NCTM.
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  • Guzmán, J., & Edilson, V. (2019). Logarithmic Equations and Applications. International Journal of Mathematics Education, 21(3), 123–134.

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